Difference Index of Quasi-regular Difference Algebraic Systems

Abstract

This paper is devoted to studying difference indices of quasi-regular difference algebraic systems. We give the definition of difference indices through a family of pseudo-Jacobian matrices. Some properties of difference indices are proved. In particular, a Jacobi-type upper bound for the sum of the order and the difference index is given. As applications, an upper bound of the Hilbert-Levin regularity and an upper bound of the order for difference ideal membership problem are deduced.